Two ships A and B leaves the some harbour at the same time in directions 152° and 142° respectively calculate the distance between the two ships when A has travelled 10km and B has travelled 12km. What is the bearing of A from B at this time
sound much as the law of cosines and sines on this one.
sketch it. first you have two sides and angle between to get opposite side. Now use the law of sines to get the last angle, Notice you have to convert that to a bearing relative ot North.
All angles are measured CW from +y-axis.
Given: HA = 10km[152o], HB = 12km[142o].
BH = 12km[142+180] = 12km[322o].
BA = BH + HA = 12[322] + 10[152]
BA = (12*sin322+10*sin152) + (12*cos322+10*cos152)i
BA = -2.69 + 0.627i = 2.76km[-76.9o] = 2.76km[283o] CW.
Bearing of A from B = 283 degrees.
The sketch of the journey was not drawn
The answer is 2.765
Why formulary was used here
I'm not sure it's cosine
Well, it seems like ships A and B are going on a little adventure! Let me calculate the distance and bearing for you.
To find the distance between the two ships, we can use the formula:
Distance = sqrt((First Ship's Distance)^2 + (Second Ship's Distance)^2 - 2(First Ship's Distance)(Second Ship's Distance) * cos(First Ship's Bearing - Second Ship's Bearing))
So, substituting the values into the formula:
Distance = sqrt((10)^2 + (12)^2 - 2(10)(12) * cos(152° - 142°))
After some calculation, the distance between the two ships is approximately 20.12 km.
Now, let's find the bearing of A from B. This can be determined by subtracting Ship B's initial bearing from Ship A's initial bearing:
Bearing of A from B = 152° - 142°
The bearing of A from B is approximately 10°.
So, after traveling 10 km and 12 km, the distance between the two ships is 20.12 km with a bearing of A from B being 10°. I hope this helps you navigate through your ship-related troubles!
To solve this problem, we can use the concept of vectors and trigonometry.
Step 1: Determine the positions of ships A and B after traveling the given distances.
- Ship A has traveled 10 km in the direction of 152°. We can find its position by creating a vector with magnitude 10 km and direction 152°.
- Ship B has traveled 12 km in the direction of 142°. We can find its position by creating a vector with magnitude 12 km and direction 142°.
Step 2: Calculate the distance between the two ships.
- To calculate the distance between two points (x1, y1) and (x2, y2) in a 2D plane, we can use the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
- In this case, we have the positions of ship A and ship B as vectors, so we need to convert them into Cartesian coordinates (x, y) to use the formula.
Step 3: Find the bearing of ship A from B.
- To find the bearing of ship A from B, we need to determine the angle between the line connecting the two ships and the reference line (e.g., North).
Let's calculate the distances and bearings step by step.
Step 1:
- Ship A's position: (10 * cos(152°), 10 * sin(152°)).
- Ship B's position: (12 * cos(142°), 12 * sin(142°)). Note: The angles must be in radians for the trigonometric functions.
Step 2:
- Distance between A and B: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
- Substitute the values of x1, y1, x2, y2 into the formula to calculate the distance.
Step 3:
- Calculate the bearing by finding the angle between the line connecting the two ships and the reference line (e.g., North).